KCC's Quizzes on AQQ256 about square roots in the digital world

My thought process…

I’m going to call K = 12345678987654321 for simplicity.

N has to be divisible by 9 since the sum of the numbers in K are divisible by 9.

Since the answer (K) ends in a 1 and it’s the square of a number, N must end in a 1 or 9 since those are the only two numbers whose square is a 1 (1 and 81).

I note that N has to lie within the bounds of 100,000,000 (which gives 17 digits as an answer), and 1,000,000,000,000 (lowest to give 18 digits), so it will be a 9 digit number.

I notice there is a similarity in K with the square of 11 (= 121) and 111 (12321) as the pattern of increasing numbers and decreasing numbers are shared.

Make a guess that the solution (N) may be made up of only the digit 1 (ie, 1111….) and since the solution must be divisible by 9, as well as be a 9 digit number, I guess the solution is…

N = 111,111,111

Plugging into N2 reveals the guess was correct!  N2 = 12345678987654321.

- Brian

My thought process…

I’m going to call K = 12345678987654321 for simplicity.

N has to be divisible by 9 since the sum of the numbers in K are divisible by 9.

Since the answer (K) ends in a 1 and it’s the square of a number, N must end in a 1 or 9 since those are the only two numbers whose square is a 1 (1 and 81).

I note that N has to lie within the bounds of 100,000,000 (which gives 17 digits as an answer), and 1,000,000,000,000 (lowest to give 18 digits), so it will be a 9 digit number.

I notice there is a similarity in K with the square of 11 (= 121) and 111 (12321) as the pattern of increasing numbers and decreasing numbers are shared.

Make a guess that the solution (N) may be made up of only the digit 1 (ie, 1111….) and since the solution must be divisible by 9, as well as be a 9 digit number, I guess the solution is…

N = 111,111,111

Plugging into N2 reveals the guess was correct!  N2 = 12345678987654321.

- Brian

Correct answer with beautiful and convincing explanation Brian!